Adverse Selection in an Efficiency Wage
Model with Heterogeneous Agents
Ricardo Azevedo Araujo
Department of Economics, University of Brasilia (UnB), Brazil
Adolfo Sachsida
Brazilian Institute for Applied Economic Research (IPEA) and IBMEC-DF, Brazil
Abstract
This paper studies efficiency wages in the presence of heterogeneous workers and
asymmetric information. It includes an incentive compatibility constraint (ICC) in the
efficiency wage model with heterogeneous workers to show that the implementation of
efficiency wages in the presence of heterogeneity faces the problem of adverse selection.
Employees with a smaller effort aversion supply a smaller level of effort than what is
optimal under perfect information due to hidden information. In this vein only a second
best solution is obtained.
Keywords: Efficiency Wages, Adverse Selection, Asymmetric Information
JEL Classification: J41, M59
Resumo
Este artigo analisa a utilização de salários de eficiência na presença de trabalhadores
heterogêneos e assimetria de informação. Através da inclusão de uma restrição de
compatibilidade de incentivos nesse modelo com agentes heterogêneos, é possível mostrar
que ocorre o problema de seleção adversa. Trabalhadores com menor aversão ao esforço
ofertam um menor nível de esforço do que aquele que é ótimo sob informação perfeita.
Assim apenas uma solução de segundo ótima é possível.
⋆
Submitted in June 2010, accepted in August 2010. We would like to thank without implicating
Emilson Silva and an anonymous referee for helpful suggestions. The usual disclaimer applies. Financial
support from the CNPq is acknowledged.
E-mail address: [email protected]
Revista EconomiA
September/December 2010
Ricardo Azevedo Araujo and Adolfo Sachsida
1. Introduction
Efficiency wages models predict that homogeneous workers may receive different
wages relying on the degree of shirking aversion of firms. In their classical article,
Shapiro and Stiglitz (1984) proposed a non-shirking condition which once satisfied
avoid workers to incur in shirking. More recently Charness and Kuhn (2005) propose
an efficiency wage model with heterogeneous workers (high productivity and low
productivity workers) to explain why firms can compress wages.
The main idea of efficiency wage models is that employers are able to set wages
up its competitive level to avoid shirking. That is, employer offers a contract to an
employee that is able to avoid shirking, this is the non-shirking condition. Since
the worker accepts the contract this implies that her utility to work hard is bigger
than the utility of incurring in shirking. In other words, the non-shirking condition
may be viewed as a participation constraint.
Accepting the non-shirking condition (NSC) as a participation constraint raises
the question about the incentive compatibility constraint. While NSC warrants
participation, it does not imply that the worker will reveal his true type. In models
with homogeneous workers it is not a problem. However, it is hardly defensible
that the work force is homogeneous. Difference in abilities, skills, and motivation
are simple examples which can be mentioned to refute homogeneity hypothesis.
While skills can be observed, motivation and abilities can not be easily observed.
However, all of them have impact over productivity.
In the literature of efficiency wages with heterogeneous workers it is usual to
assume different types of individuals with particular productivities. Furthermore,
it is implicit assumed that the identification of the productivity level of the worker
is easy. Since the employer can always verify the worker productivity – and classify
him/her as low or high productive – the incentive compatibility constraint is not
necessary.
Facing a heterogeneous supply of labor, employers have some difficulties to
learn about employee’s productivity, mainly during the hiring process. In this
scenario, the incentive compatibility constraint (ICC) becomes important and a
device to force workers to reveal their true types becomes necessary. In order
to tackle this point this paper extends the Shapiro-Stiglitz model to allow for
heterogeneity between workers. 1 Assuming that the heterogeneity comes from the
disutility to work, we introduce an ICC condition in the efficiency wage model.
Our main result shows that the implementation of efficiency wages in the presence
of heterogeneous workers gives rise only to a second best solution due to the issue
of hidden information: workers with a smaller disutility of effort act as if they
had a higher aversion to effort in order to obtain a better contract. This paper is
structured as follows: Section 3 presents the formal model and Section 4 concludes.
1
Shapiro-Stiglitz original paper suggests that heterogeneous workers could be a source of stigma from
loosing a job. This stigma could serve as a discipline device. However, they do not deal with the
implications of the heterogeneity of workers over wages which is the objective of this paper.
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Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents
2. The Model
Let us assume an one period model in which the utility function is given by:
u(w, e) = w − Re, where w is the wage paid by the firm, e is the effort, and R is
a positive constant that measures the disutility of effort (inverse of motivation). 2
The employee has outside opportunities that provide him with a reservation salary
w̄ with no effort (e = 0). There is a probability p to lose the job attached to shirking
behavior. The expected utility is given by:
VN = w − Re
(1)
VS = pw̄ + (1 − p)w
(2)
where VN is the utility associated with no shirking and Vs the utility associated
with shirking behavior. The efficiency wage is chosen to satisfy the inequality:
(3)
VN ≥ VS
Due to the maximization behavior of the firm the equality holds, that is:
w − Re = pw̄ + (1 − p)w
(4)
Implicit derivation of w as a function of R yields:
Re
(5)
p
This seems that wage may be an increasing function of R. Let us specify the
contract under perfect information, which is assumed to be given by:
w=w+
M axF [e] − w
w,e
s.t. w − Re ≥ pw̄ + (1 − p)w
(6)
(NSC)
where the price of the product is normalized to one, and F (.) is a production
function, with F ′ (.) > 0 and F ”(.) < 0. From the constraint we obtain: w = w̄+ Re
p .
w̄)
. By inserting this expression into
This expression may be rewritten as e = p(w−
R
the objective function and deriving with respect to w we obtain: F ′ (.) = R
p . This
expression shows that the marginal productivity of effort is an increasing function
of the disutility of effort (R). By assuming that: F [e] = eα then after some algebraic
manipulation the expression above yields the following efficiency wage:
1
w = w̄ + α 1−α
α
p 1−α
R
(7)
2
Katz (1986), Akerlof and Yellen (1990), and Charness and Kuhn (2005) also used an effort that is a
function of wages.
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∂w
By taking the partial derivative of w with respect to R we conclude that: ∂R
=
1
α
1−α
p 1−α
−α R
< 0. This result shows that the optimal strategy for the firm is to
R
set wages to make it a decreasing function of R. Let us assume now that there are
only two types of workers: 3 low aversion to effort, R1 , and high aversion to effort,
R2 , with R2 >R1 . The incentive compatibility constraint for worker type 1 requires
that:
w 1 − R 1 e1 ≥ w 2 − R 1 e2
(IC1)
where w1 and w2 stand for the wages for the workers of type 1 and 2, respectively
and IC1 stands for the incentive compatibility constraint for worker of type 1.
Expression (IC1) requires that the utility to behave like type 1, when the worker
is type 1, should be larger (or at least equal) than the utility to behave like type
2. In the same way, there is an incentive compatibility restriction to worker type
2, namely (IC2), which is written as:
w 2 − R 2 e2 ≥ w 1 − R 2 e1
(IC2)
The non-shirking conditions for workers 1 and 2, namely (NS1) and (NS2)
respectively, require that:
w1 − R1 e1 ≥ pw̄ + (1 − p)w
(NS1)
w2 − R2 e2 ≥ pw̄ + (1 − p)w
(NS2)
They state that each worker prefers the contract that is designed for her. When
the firm is hiring the worker or proposing the contract it does not know the type
of the employee. Let us assume that the only information the firm has is the
distribution of workers of type 1 and 2 in the population and that the probability
of hiring a worker of type 1 is σ, and the remainder, 1 − σ, is the probability of
hiring a worker of type 2. The contract under imperfect information that satisfies
the revelation principle is given by:
M ax
w1 ,w2 ,e1 ,e2
σ[F (e1 ) − w1 ] + (1 − σ)[F (e2 ) − w2 ]
(8)
s.t. w1 − R1 e1 ≥ w2 − R1 e2
(IC1)
w 2 − R 2 e2 ≥ w 1 − R 2 e1
(IC2)
w1 − R1 e1 ≥ pw̄ + (1 − p)w
(NS1)
3
The number of types could be extended to consider $n$ different types but the case in which there
are two types in enough to convey the main implications of the model without cumbersome algebraic
manipulations.
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Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents
w2 − R2 e2 ≥ pw̄ + (1 − p)w
(NS2)
A possible way of tackling the above problem is to adopt the Kuhn-Tucker
Theorem. Before doing this let us show that one of the restrictions of the problem,
namely restriction (NS1), may be ignored. This is the content of the following:
Lemma 1: The constraint (NS1) may be ignored in problem (8).
Proof. From (IC1) from the fact that R1 <R2 one obtains: w1 −R1 e1 ≥ w2 −R1 e2 >
w2 − R2 e2 . From (NS2) it is possible to conclude that: w1 − R1 e1 ≥ pw̄ + (1 − p)w.
Another useful Lemma is the one that shows that (NS2) holds with equality.
Lemma 2: (NS2) holds with equality.
Proof. Assume for the sake of contradiction that the optimal contract is the one
in which w2 − R2 e2 > pw̄ + (1 − p)w. Then the firm may choose another contract
by choosing w2 − ǫ that still satisfies the above equations and yields smaller costs.
But this is a contradiction to fact that the first contract was optimal. This result may also be proven by solving the maximization problem (8) subject
to (IC1), (IC2) and (NS2) through the Kuhn-Tucker method. By using Lemma 1,
the problem may be rewritten as:
M ax
w1 ,w2 ,e1 ,e2
(9)
σ[F (e1 ) − w1 ] + (1 − σ)[F (e2 ) − w2 ]
s.t. w1 − R1 e1 ≥ w2 − R1 e2
(IC1)
w 2 − R 2 e2 ≥ w 1 − R 2 e1
(IC2)
w2 − R2 e2 ≥ pw̄ + (1 − p)w
(NS2)
The Lagrangean function related to this problem may be written as:
L = σ[F (e1 ) − w1 ] + (1 − σ)[F (e2 ) − w2 ] + λ1 [w1 − R1 e1 − w2
+ R1 e2 ] + λ2 [w2 − R2 e2 − w1 + R2 e1 ] + µ [w2 − R2 e2 − pw̄ − (1 − p)w]
where λ1 , λ2 and ì are the Kuhn-Tucker multipliers related to constraints (IC1),
(IC2) and (NS2) respectively. The first order conditions are given by:
∂L
= −σ + λ1 − λ2 = 0
∂w1
(A1)
∂L
= −(1 − σ) − λ1 + λ2 + µ = 0
∂w2
(A2)
∂L
= σF ′ (e1 ) + λ1 R1 − λ2 R2 = 0
∂e1
(A3)
∂L
= (1 − σ)F ′ (e2 ) − λ1 R1 + λ2 R2 + µR2 = 0
∂e2
(A4)
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The Kuhn-Tucker conditions are given by:
λ1
∂L
∂L
= λ1 [w1 − R1 e1 − w2 + R1 e2 ] = 0, λ1 ≥ 0,
= w 1 − R 1 e1 − w 2 + R 1 e2 ≥ 0
∂λ1
∂λ1
(A5)
λ2
∂L
∂L
= λ2 [w2 − R2 e2 − w1 + R2 e1 ] = 0, λ2 ≥ 0,
= w 2 − R 2 e2 − w 1 + R 2 e1 ≥ 0
∂λ2
∂λ2
(A6)
∂L
∂L
= µ [w2 − R2 e2 − pw̄ − (1 − p)w] = 0, µ ≥ 0,
= w2 −R2 e2 −pw̄−(1−p)w ≥ 0
∂µ
∂µ
(A7)
It is possible to show – see the Appendix – that from the eight possible cases
just one is feasible, namely case (v), in which λ1 = 0, λ2 = σ > 0 and µ = 1 > 0.
In this context the following propositions hold.
Proposition 1: The utility of a worker of type 1 is larger than the utility of a
worker of type 2, that is: w1 − R1 e1 > w2 − R2 e2
Proof: From (IC1) we know that: w1 −R1 e1 ≥ w2 −R1 e2 . Since R1 <R2 we conclude
that: w1 − R1 e1 ≥ w2 − R1 e2 > w2 − R2 e2 . This proposition shows that that the firm makes the utility of an employee of
type 1, with higher willingness to work to be larger than the utility of an employee
of type 2. At a first glance the contract provides right incentives since it gives more
rewards in terms of utility to workers with higher willingness to work. Proposition
2 shows that the effort of a worker of type 1 is chosen to be larger or at least equal
than the effort of a worker of type 2.
Proposition 2: The effort of a worker of type 1 is larger or equal than the wage
of a worker of type 2. That is: e1 ≥ e2 .
Proof: By summing (IC1) and (IC2) and after some algebraic manipulation we
obtain: (R1 − R2 )[e2 − e1 ] ≥ 0. Since R1 <R2 we conclude that: e2 ≤ e1 . Hence the
effort of a worker of type 1 is larger or equal than the salary of a worker of type 2.
From Propositions 1 and 2 it is possible to conclude that in general a worker
of type 1 has better reward than a worker of type 2. This result is according to
what was established as the wage chosen by the firm as the outcome of the profit
maximization behavior of the firm under perfect information and it shows that the
conventional wisdom that workers with higher willingness to work have to receive
larger wages than those which higher effort aversion prevails. However since the
worker of type 1 has to provide an effort level higher or at least equal to the one
provided by worker or type 2 it may find wrong incentives to announce that she
is a worker of type 2 in order to provide less effort than what it is optimal under
perfect information.
Proposition 3: An employee of type 1 chooses a contract designed for an employee
of type 2.
µ
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Proof: From Proposition 2, we know that: e2 ≤ e1 . From (IC2) we know that
w1 − R1 e1 ≥ w2 − R1 e2 . Note that it is not possible to exclude the possibility that:
w1 − R1 e1 = w2 − R1 e2 . In fact this result holds as it is shown in case (vi) of the
Appendix. In this vein a worker of type 1 by announcing of being a worker of type
2 may provide smaller effort and receives the same utility as if he admitted his own
type. By announcing to be a worker of type 2 allows the worker of type 1 to provide less
effort while keeping her utility of being a worker of type 1. The consequence of this
strategy is that the salary earned by her will be smaller than if she admitted her
true type. Of course this fall in wages will be compensated in the utility function
by the smaller effort provided. But a further analysis on this outcome shows that
it can produce only a second best result for the firm: profits that the firm obtains
hiring an employee of type 1 is smaller than what it would obtain under perfect
information. This is the content of the next proposition.
Proposition 4: Profits of the firm hiring worker of type 1 are smaller than profits
it obtains under perfect information.
Proof: From the profit maximization problem of the firm it is possible to conclude
that the profit of the firm under perfect information by hiring a worker of type 1
is: Π1 = F [e1 ] − w1 where w1 = w̄ + R1pe1 . This yields: Π1 = F [e1 ] − w̄ − R1pe1 .
The same result holds for the profit of the firm by hiring a worker of type 2 under
perfect information, that is: Π2 = F [e2 ] − w̄ − R2pe2 . It is possible to show that
the profit of the firm by hiring a worker of type 1 is large or at least equal to the
profit of the firm hiring a worker of type 2: as R1 <R2 then F [e2 ] − w̄ − R1pe2 >
F [e2 ]− w̄− R2pe2 = Π2 . But we know that Π1 = F [e1 ]− w̄− R1pe1 ≥ F [e2 ]− w̄− R1pe2
since e1 is the solution for the profit maximization under perfect information if the
effort aversion is R1 . Hence we conclude that Π1 > Π2 . Note that the common
term in both inequalities is nothing but the profit of the firm by hiring a worker of
type 1, acting as worker of type 2, which is smaller than the profit of a worker of
type 1 under perfect information and higher than the profit of a worker of type 2
under perfect information. This result is not the first best since the effort provided by a worker of type 1
is smaller than her effort under perfect information which leads to smaller profits.
Hence the contract is not optimal since it produces only a second best solution.
The model shows that after the contract is signed and productivity is revealed it
is possible to identify workers of type 1 and 2 by the efforts they provide. Hence
a possible way to avoid the second best solution due to hidden information is to
establish a probation period in which the productivity of workers is screened. In this
case, workers of type 1, by knowing that they are identified during the probation
period find right incentives to announce his true productivity when the contract is
signed. But it is important to bear in mind that this device works only in the case
in which the probation period does not create incentives to workers of type 1 to
remain only during this period and quit the job after that and that they are aware
of their own productivity.
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Pouyet et al. (2008) assuming that principals compete for attracting
heterogeneous agents by offering contracts have found that when the agents’ types
are publicly observed then competitive equilibria are efficient, a result similar to
the one obtained here. When types are privately observed these authors continue
to obtain the result that efficiency holds provided that principals do not directly
care about the agents’ private information. Here we have shown that only a second
best solution is possible in the case of hidden information.
3. Concluding Remarks
In this paper we get apart from homogeneous work hypothesis and derive
an incentive compatibility constraint (ICC) for the efficiency wage approach.
Supposing that the heterogeneity arises from different disutility to work
(motivation) between workers, we show that the efficiency wage contract provides
only second best solution: employee of type 1 supplies a smaller level of effort than
what is optimal under perfect information due to hidden information.
References
Akerlof, G. & Yellen, J. (1990). The fair-wage effort hypothesis and unemployment.
Quarterly Journal of Economics, 105(2):255–284.
Charness, G. & Kuhn, P. (2005). Pay inequality, pay secrecy and effort: Theory
and evidence. NBER Working Paper 11786.
Katz, L. (1986). Efficiency wage theories: A partial evaluation. In NBER
Macroeconomics Annual, pages 235–76. NBER.
Pouyet, J., Salanié, B., & Salanié, F. (2008). On competitive equilibria with
asymmetric information. The B.E. Journal of Theoretical Economics, 8(1).
Available at: http://www.bepress.com/bejte/vol8/iss1/art13.
Shapiro, C. & Stiglitz, J. E. (1984). Equilibrium unemployment as a worker
discipline device. American Economic Review, 74(3):433–44.
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Appendix
Let us analyze the eight possible cases, which arise from the possible combination
of the Kuhn-Tucker multipliers:
(i) λ1 = λ2 = µ = 0. From expression (A1) it yields that σ = 0, a contradiction.
(ii) λ1 = λ2 = 0 and µ > 0. For the same reason of case (i) we exclude this
possibility.
(iii) λ1 = µ = 0 and λ2 > 0. From expression (A1) it yields λ2 = −σ < 0, a
contradiction.
(iv) λ2 = µ = 0 and λ1 > 0. From expression (A1) it yields λ1 = σ. By
substituting this result into expression (A2) we conclude that µ = 1, a
contradiction.
(v) λ1 = 0, λ2 > 0and µ > 0. From expression (A1) it yields λ2 = −σ < 0, a
contradiction.
(vi) λ1 > 0, λ2 = 0 and µ > 0. From expression (A1) it yields λ1 = σ > 0.
By substituting this result into expression (A2) we conclude that µ = 1.
Note that in this case, from expression (A6) it is possible to conclude that
w2 −R2 e(w2 ) = w1 −R2 e(w1 ), which is the content of Proposition 3, meaning
that a worker of type 1 is indifferent between her contract and a contract for
the type 1. Besides as µ = 1 > 0 then (NS2) holds with equality which is the
content of Lemma 2.
(vii) λ1 > 0, λ2 > 0 and µ = 0. By substituting (A1) into (A2) we conclude that
µ = 1, a contradiction to the fact that µ = 0.
(viii) λ1 > 0, λ2 > 0 and µ > 0. From (A1) λ1 − λ2 = σ. By inserting this
expression into expression (A2) one obtains µ = 1. As all the multipliers are
different from zero then the constraints are binding. By summing up (A3)
and (A4) and after some algebraic manipulation we conclude that: σF ′ (e1 ) +
(1 − σ)F ′ (e2 ) + R2 = 0. All the terms in the left hand side of this equality
are larger than zero then it is not possible to hold.
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